Application of the Parallel Dichotomy Algorithm for solving Toeplitz tridiagonal systems of linear equations with one right-hand side
It addresses the need for efficient parallel solvers for Toeplitz tridiagonal systems, which are common in numerical methods, but the improvement is incremental.
The paper proposes a parallel modification of the Dichotomy Algorithm for solving Toeplitz tridiagonal systems with one right-hand side, achieving nearly linear speedup and accuracy comparable to the Thomas method.
Basing on a modification of the "Dichotomy Algorithm" (Terekhov, 2010), we propose a parallel procedure for solving tridiagonal systems of equations with Toeplitz matrices. Taking the structure of the Toeplitz matrices, we may substantially reduce the number of the "preliminary calculations" of the Dichotomy Algorithm, which makes it possible to effectively solve a series as well as a single system of equations. On the example of solving of elliptic equations by the Separation Variable Method, we show that the computation accuracy is comparable with the sequential version of the Thomas method, and the dependence of the speedup on the number of processors is almost linear. The proposed modification is aimed at parallel realization of a broad class of numerical methods including the inversion of Toeplitz and quasi-Toeplitz tridiagonal matrices.